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# An application of the inequality for modified Poisson kernel

*Journal of Inequalities and Applications*
**volumeÂ 2016**, ArticleÂ number:Â 24 (2016)

## Abstract

As an application of an inequality for modified Poisson kernel obtained by Qiao and Deng (Bull. Malays. Math. Sci. Soc. (2) 36(2):511-523, 2013), we give the generalized solution of the Dirichlet problem with arbitrary growth data.

## 1 Introduction and results

Let \(\mathbf{R}^{n}\) (\(n\geq2\)) be the *n*-dimensional Euclidean space. The boundary and the closure of a set *E* in \(\mathbf{R}^{n}\) are denoted by *âˆ‚E* and *EÌ…*, respectively. The Euclidean distance of two points *P* and *Q* in \(\mathbf{R}^{n}\) is denoted by \(\vert P-Q\vert \). Especially, \(\vert P\vert \) denotes the distance of two points *P* and *O* in \(\mathbf{R}^{n}\), where *O* is the origin in \(\mathbf{R}^{n}\).

We introduce a system of spherical coordinates \((r,\Theta)\), \(\Theta=(\theta_{1},\theta_{2},\ldots,\theta_{n-1})\), in \(\mathbf{R}^{n}\) which are related to Cartesian coordinates \((x_{1},x_{2},\ldots,x_{n-1},x_{n})\) by \(x_{n}=r\cos\theta_{1}\).

Let \(B(P,r)\) denote the open ball with center at *P* and radius *r* (>0) in \(\mathbf{R}^{n}\). The unit sphere and the upper half unit sphere in \(\mathbf{R}^{n} \) are denoted by \(\mathbf{S}^{n-1}\) and \(\mathbf{S}_{+}^{n-1}\), respectively. The surface area \(2\pi^{n/2}\{\Gamma(n/2)\}^{-1}\) of \(\mathbf{S}^{n-1}\) is denoted \(w_{n}\). Let \(\Omega\subset\mathbf{S}^{n-1}\), a point \((1,\Theta)\) and the set \(\{\Theta; (1,\Theta)\in\Omega\}\) are denoted Î˜ and Î©, respectively. For two sets \(\Lambda\subset\mathbf{R}_{+}\) and \(\Omega\subset\mathbf{S}^{n-1}\), we denote \(\Lambda\times\Omega=\{ (r,\Theta)\in\mathbf{R}^{n}; r\in\Lambda,(1,\Theta)\in\Omega\}\), where \(\mathbf{R}_{+}\) is the set of all positive real numbers.

For the set \(\Omega\subset\mathbf{S}^{n-1}\), we denote the set \(\mathbf{ R}_{+}\times\Omega\) in \(\mathbf{R}^{n}\) by \(C_{n}(\Omega)\), which is called a cone. For the set \(I\subset \mathbf{R}\), the sets \(I\times\Omega\) and \(I\times\partial{\Omega}\) are denoted \(C_{n}(\Omega;I)\) and \(S_{n}(\Omega;I)\), respectively, where **R** is the set of all real numbers. Especially, the set \(S_{n}(\Omega; \mathbf{R}_{+})\) is denoted \(S_{n}(\Omega)\).

Given a continuous function *f* on \(S_{n}(\Omega)\), we say that *h* is a solution of the Dirichlet problem in \(C_{n}(\Omega)\) with *f*, if *h* is a harmonic function in \(C_{n}(\Omega)\) and

Let \(\Omega\subset\mathbf{S}^{n-1}\) and \(\Delta^{*}\) be a Laplace-Beltrami on the unit sphere. Consider the Dirichlet problem (see, *e.g.* [2], p.41)

We denote the non-decreasing sequence of positive eigenvalues of it, repeating accordingly to their multiplicities, and the corresponding eigenfunctions are denoted, respectively, by \(\{\lambda_{i}\}_{i=1}^{\infty}\) and \(\{\varphi_{i}(\Theta)\}_{i=1}^{\infty}\). Especially, we denote the least positive eigenvalue of it \(\lambda_{1}\) and the normalized positive eigenfunction to \(\lambda_{1}\)
\(\varphi_{1}(\Theta)\). In the sequel, for the sake of brevity, we shall write *Î»* and *Ï†* instead of \(\lambda_{1}\) and \(\varphi_{1}\), respectively.

The set of sequential eigenfunctions corresponding to the same value of \(\{\lambda_{i}\}_{i=1}^{\infty}\) in the sequence \(\{\varphi_{i}(\Theta)\}_{i=1}^{\infty}\) makes an orthonormal basis for the eigenspace of the eigenvalue \(\lambda_{i}\). Hence for each \(\Omega\subset S^{n-1}\) there is a sequence \(\{k_{j}\}\) of positive integers such that \(k_{1}=1\), \(\lambda_{k_{j}}<\lambda_{k_{j+1}}\), \(\lambda_{k_{j}}=\lambda_{k_{j}+1}=\lambda_{k_{j}+2}=\cdots=\lambda_{k_{j+1}-1}\) and \(\{\varphi_{k_{j}},\varphi_{k_{j}+1},\ldots,\varphi_{k_{j+1}-1}\}\) is an orthonormal basis for the eigenspace of the eigenvalue \(\{\lambda_{k_{j}}\}_{j=1}^{\infty}\). By \(I_{\Omega}(k_{m})\) we denote the set of all positive integers less than \(\{k_{m}\}_{m=1}^{\infty}\). In spite of the fact

the summation over \(I_{\Omega}(k_{1})\) of a function \(S(k)\) of a variable *k* will be used by promising

If we denote the solutions of the equation

by \(\aleph_{i}^{+}\) and \(\aleph_{i}^{-}\), then the functions

are harmonic functions in \(C_{n}(\Omega)\) and vanish on \(S_{n}(\Omega)\).

Let \(G_{\Omega}(P,Q)\) be the Green function of \(C_{n}(\Omega)\) for any \(P=(r,\Theta)\in C_{n}(\Omega)\) and any \(Q=(t,\Phi)\in C_{n}(\Omega)\). Then the Poisson kernel in \(C_{n}(\Omega)\) can be defined by

where \(P\in C_{n}(\Omega)\), \(Q\in S_{n}(\Omega)\), \({\partial}/{\partial n_{Q}}\) denotes the differentiation at *Q* along the inward normal into \(C_{n}(\Omega)\) and

Let \(F(\Theta)\) be a function defined in Î©. We denote \(N_{i}(F)\) by

when it exists.

For any two points \(P=(r,\Theta) \) and \(Q=(t,\Phi)\) in \(C_{n}(\Omega)\) and \(S_{n}(\Omega)\), respectively, we define

where *m* is a non-negative integer and

To obtain the solution of the Dirichlet problem in a cone, as in [1, 3, 4], we use the modified Poisson kernel defined by

where \(P\in C_{n}(\Omega)\) and \(Q\in S_{n}(\Omega)\), which has the following estimates (see [1]):

for any \(P=(r,\Theta)\in C_{n}(\Omega)\) and any \(Q=(t,\Phi)\in S_{n}(\Omega)\) satisfying \(0<\frac{r}{t}<\frac{1}{2}\), where *M* is a constant independent of *P*, *Q*, and *m*. For the construction and applications of a modified Green function in a half space, we refer the reader to the paper by Qiao (see [5]).

Write

where \(f(Q)\) is a continuous function on \(\partial C_{n}(\Omega)\) and \(d\sigma_{Q}\) the \((n-1)\)-dimensional volume elements induced by the Euclidean metric on \(\partial{C_{n}(\Omega)}\).

Recently, Qiao and Deng (*cf.* [1]) gave the solution of the Dirichlet problem in a cone. Applications of modified Poisson kernel with respect to the SchrÃ¶dinger operator, we refer the reader to the papers by Huang and Ychussie (see [6]) and Li and Ychussie (see [7]).

### Theorem A

*If*
\(\Omega+\aleph^{+}-1>0\), \(\Omega-n+1\leq\aleph_{k_{m+1}}^{+}<\Omega-n+2\)
*and*
\(f(Q)\) (\(Q=(t,\Phi )\)) *is a continuous function on*
\(\partial{C_{n}(\Omega)}\)
*satisfying*

*then the function*
\(U_{\Omega}^{m}[f](P)\)
*is a solution of the Dirichlet problem in*
\(C_{n}(\Omega)\)
*with*
*f*
*and*

Furthermore, Qiao and Deng (*cf.* [4]) supplemented the above result and proved the following.

### Theorem B

*Let*
\(0< p<\infty\), \(\gamma>(-\aleph^{+}-n+2)p+n-1\)
*and*

*If*
\(f(Q)\) (\(Q=(t,\Phi)\)) *is a continuous function on*
\(S_{n}(\Omega)\)
*satisfying*

*then the function*
\(U_{\Omega}^{m}[f](P)\)
*satisfies*

It is natural to ask if the continuous function *u* satisfying (2) and (3) can be replaced by arbitrary continuous function? In this paper, we shall give an affirmative answer to this question. To do this, we first construct a modified Poisson kernel. Let \(\phi(l)\) be a positive function of \(l\geq1\) satisfying

Denote the set

by \(\pi_{\Omega}(\phi,i)\). Then \(1\in\pi_{\Omega}(\phi,i)\). When there is an integer *N* such that \(\pi_{\Omega}(\phi,N)\neq\Phi\) and \(\pi_{\Omega}(\phi,N+1)= \Phi \), denote

of integers. Otherwise, denote the set of all positive integers by \(J_{\Omega}(\phi)\). Let \(l(i)=l_{\Omega}(\phi,i+1)\) be the minimum elements *l* in \(\pi _{\Omega}(\phi,i)\) for each \(i\in J_{\Omega}(\phi)\). In the former case, we put \(l{(N+1)}=\infty\). Then \(l(1)=1\). The kernel function \(\widetilde{K}_{\Omega}^{\phi}(P,Q)\) is defined by

where \(P\in C_{n}(\Omega)\) and \(Q=(t,\Phi)\in S_{n}(\Omega)\).

The generalized Poisson kernel \(P_{\Omega}^{\phi}(P,Q)\) is defined by

where \(P\in C_{n}(\Omega)\) and \(Q\in S_{n}(\Omega)\).

As an application of the inequality (1) and the generalized Poisson kernel \(PI_{\Omega}^{\phi}(P,Q)\), we have the following.

### Theorem

*Let*
\(g(Q)\)
*be a continuous function on*
\(S_{n}(\Omega)\). *Then there is a positive continuous function*
\(\phi_{g}(l)\)
*of*
\(l\geq0\)
*depending on*
*g*
*such that*

*is a solution of the Dirichlet problem in*
\(C_{n}(\Omega)\)
*with*
*g*.

## 2 Lemmas

### Lemma 1

*Let*
\(\phi(l)\)
*be a positive continuous function of*
\(l\geq1\)
*satisfying*

*Then*

*for any*
\(P=(r,\Theta)\in C_{n}(\Omega)\)
*and any*
\(Q=(t,\Phi)\in S_{n}(\Omega)\)
*satisfying*

### Proof

We can choose two points \(P=(r,\Theta)\in C_{n}(\Omega)\) and \(Q=(t,\Phi)\in S_{n}(\Omega)\), which satisfies (4). Moreover, we also can choose an integer \(i=i(P,Q)\in J_{\Omega}(\phi)\) such that

Then

Hence we have from (1), (4), and (5) that

which is the conclusion.â€ƒâ–¡

### Lemma 2

(See [4])

*Let*
\(g(Q)\)
*be a continuous function on*
\(\partial{C_{n}(\Omega)}\)
*and*
\(V(P,Q)\)
*be a locally integrable function on*
\(\partial{C_{n}(\Omega)}\)
*for any fixed*
\(P\in C_{n}(\Omega)\), *where*
\(Q\in \partial{C_{n}(\Omega)}\). *Define*

*for any*
\(P\in C_{n}(\Omega)\)
*and any*
\(Q\in\partial{C_{n}(\Omega)}\).

Suppose that the following two conditions are satisfied:

(I) For any \(Q'\in\partial{C_{n}(\Omega)}\) and any \(\epsilon>0\), there exists a neighborhood \(B(Q')\) of \(Q'\) such that

for any \(P=(r,\Theta)\in C_{n}(\Omega)\cap B(Q')\), where *R* is a positive real number.

(II) For any \(Q'\in\partial{C_{n}(\Omega)}\), we have

for any positive real number *R*.

Then

for any \(Q'\in\partial{C_{n}(\Omega)}\).

## 3 Proof of Theorem

Take a positive continuous function \(\phi(l)\) (\(l\geq1\)) such that

and

for \(l>1\), where

For any fixed \(P=(r,\Theta)\in C_{n}(\Omega)\), we can choose a number *R* satisfying \(R>\max\{1,4r\}\). Then we see from LemmaÂ 1 that

Obviously, we have

which gives

To see that \(H_{\Omega}^{\phi_{g}}(P)\) is a harmonic function in \(C_{n}(\Omega)\), we remark that \(H_{\Omega}^{\phi_{g}}(P)\) satisfies the locally mean-valued property by Fubiniâ€™s theorem.

Finally we shall show that

for any \(Q'=(t',\Phi')\in \partial{C_{n}(\Omega)}\). Set

in LemmaÂ 2, which is locally integrable on \(S_{n}(\Omega)\) for any fixed \(P\in C_{n}(\Omega)\). Then we apply LemmaÂ 2 to \(g(Q)\) and \(-g(Q)\).

For any \(\epsilon>0\) and a positive number *Î´*, by (9) we can choose a number *R* (\(>\max\{1,2(t'+\delta)\}\)) such that (6) holds, where \(P\in C_{n}(\Omega)\cap B(Q',\delta)\).

Since

as \(P=(r,\Theta)\rightarrow Q'=(t',\Phi')\in S_{n}(\Omega)\), we have

where \(Q\in S_{n}(\Omega)\) and \(Q'\in S_{n}(\Omega)\). Then (7) holds.

Thus we complete the proof of Theorem.

## References

Qiao, L, Deng, GT: Growth property and integral representation of harmonic functions in a cone. Bull. Malays. Math. Soc.

**36**(2), 511-523 (2013)Rosenblum, G, Solomyak, M, Shubin, M: Spectral Theory of Differential Operators. VINITI, Moscow (1989)

Qiao, L: Integral representations for harmonic functions of infinite order in a cone. Results Math.

**61**(1-2), 63-74 (2012)Qiao, L: Growth of certain harmonic functions in an

*n*-dimensional cone. Front. Math. China**8**(4), 891-905 (2013)Qiao, L: Modified Poisson integral and Green potential on a half-space. Abstr. Appl. Anal.,

**2012**, Article ID 765965 (2012)Huang, J, Ychussie, B: The modification of Poisson-Sch integral on cones and its applications. Filomat (to appear)

Li, Z, Ychussie, B: Sharp geometrical properties of a-rarefied sets via fixed point index for the SchrÃ¶dinger operator equations. Fixed Point Theory Appl.

**2015**, 89 (2015)

## Acknowledgements

The authors are very thankful to the anonymous referees for their valuable comments and constructive suggestions.

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All authors contributed equally to the manuscript and read and approved the final manuscript.

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Xue, G., Wang, J. An application of the inequality for modified Poisson kernel.
*J Inequal Appl* **2016**, 24 (2016). https://doi.org/10.1186/s13660-016-0959-6

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DOI: https://doi.org/10.1186/s13660-016-0959-6